where \(C^{ab}\)  are the Lorentz ghosts and \(\omega _\mu {}^{ab}\)  is the spin connection constructed of the vierbein and the gravitino.E.g., total forms
\(\omega (C,\xi ,{\cal T})\)  that correspond to the first and second term in (REF ) are
\(\Xi W_{\alpha \beta \gamma }\xi ^\alpha G^{\beta \gamma }\)  and
\(\Xi C W_{\alpha \beta \gamma }W^{\alpha \beta \gamma }\) , respectively,
where \(\Xi =(1/24)\epsilon _{abcd}\xi ^a\xi ^b\xi ^c\xi ^d\) , and \(G^{\beta \gamma }\)  and \(C\)  correspond to \(\partial _\rho A_\sigma -\partial _\sigma A_\rho \)  and \(C\) , respectively.
This can be shown by means of contracting homotopies which eliminate “trivial pairs” from the cohomology of \(s\)  and by means of the properties of the Koszul-Tate differential contained in \(s\)  which allow to remove from the antifield independent part of a cocylce of \(s\)  all supercovariant tensors (such as the supercovariantized Ricci tensor or the auxiliary fields) which vanish on-shell or are equal on-shell to polynomials in the \({\cal T}\) 's, cf., e.g., [1]}.
